\(\mathbf {(II)}\)  To show that all the unstable eigenvalues of the pole pencil belonging to realization (REF ) are poles of \(\mathbf {G}_1\mathbf {G}_2\) , we now show that they are also unstable eigenvalues of the pole pencil belonging to the realization's minimal part. We therefore separate, for the realization from (REF ), first its observable part followed by the controllable part of the latter partition (see section 3.3 of [1]}) to obtain the partitioned realization
\(\mathbf {G}_1(\lambda )\mathbf {G}_2(\lambda )=\hspace{-2.84526pt}\left[\footnotesize \begin{array}{ccc|c}\hspace{-5.69054pt}A_{11}-\lambda I_{n_{11}}&A_{12}&A_{13}&B_{11}\\O&\hspace{-8.53581pt}A_{22}-\lambda I_{n_{22}}&A_{23}&B_{21}\\O&O&\hspace{-8.53581pt}A_{33}-\lambda I_{n_{33}}&O\\\hline O&C_{12}&C_{13}&D\end{array}\right].\) 
